The thing is, an SF MP can't vote or debate because they, by policy, refuse to attend. You're correct that the party has representatives in the other bodies which it does recognise but those aren't the same people.
So for example Órfhlaith Begley was elected MP for West Tyrone. Her voters will have known she's not going to Westminster, and she didn't - but she's not in some parallel institution instead, AFAIK there isn't one. Nicola Brogan represents West Tyrone as an MLA in Stormont, because Sinn Féin does recognise the Assembly and you can't say well but there's a body in Dublin. Dublin doesn't control West Tyrone so what would she even do there?
You're absolutely right that they aren't in a parallel institution. As I understand it, the SF perspective is that there is no legitimate government representing the people of West Tyrone (other than Stormont, which is complicated). I think it sounds like we're saying the same thing but there's a subtle interpretation difference. When Idi Amin declared himself to be the King of Scotland the Scots didn't send representatives to Uganda to form a government. The SF position since their Árd Fheis before the 1919 elect has been to treat Westminster like Idi Amin. To simply engage in a conversation about Idi Amin's claim to the throne of Scotland would be to give him too much credibility.
So the core issue, from the Republican perspective, is that the people of West Tyrone are denied representation at the national level in Ireland by a foreign government. Once that representation is achieved the SF representatives will participate in it. (again, not trying to address the merits; just clarify the logic)
But Órfhlaith Begley does go to Westminster, and she has an access badge for the Houses of Parliament, and she flies over there and works in her office in the parliament buildings and answers her @parliament.co.uk emails and asks her staffers (paid for by the parliament) to respond to letters on paper headed Órfhlaith Begley MP. All the work of an MP except the most performative part.
She just doesn't go into the room which is called the House of Commons and try to speak or vote there, because the armed guards at the door won't let in anyone who doesn't swear allegiance to the King. If swearing allegiance to the King was a requirement to use the email system, then she wouldn't do that either.
Fascinating, I didn't know that they actually physically travel to Westminster. That does seem like quite a trip given they could work from home. I'm also not sure how effective you can be in this way, but of course it's really up to her constituents not me whether they feel adequately represented.
I think it's parliament.uk rather than parliament.co.uk by the way.